# inte^x[((sin^-1x)sqrt(1-x^2)+1)/sqrt(1-x^2)]dx ?

Jul 28, 2016

${e}^{x} \left({\sin}^{-} 1 x\right) + C$

#### Explanation:

We have:

$\int {e}^{x} \left[\frac{\left({\sin}^{-} 1 x\right) \sqrt{1 - {x}^{2}} + 1}{\sqrt{1 - {x}^{2}}}\right] \mathrm{dx}$

Splitting up the fraction, this becomes:

$= \int {e}^{x} \left[{\sin}^{-} 1 x + \frac{1}{\sqrt{1 - {x}^{2}}}\right] \mathrm{dx}$

Split up the integrals:

$= \int {e}^{x} \left({\sin}^{-} 1 x\right) \mathrm{dx} + \int {e}^{x} / \sqrt{1 - {x}^{2}} \mathrm{dx}$

Attempt to integrate $\int {e}^{x} \left({\sin}^{-} 1 x\right) \mathrm{dx}$ via integration by parts, which takes the form $\int u v = u v - \int v \mathrm{du}$.

Let $u = {\sin}^{-} 1 x$, so $\mathrm{du} = \frac{1}{\sqrt{1 - {x}^{2}}} \mathrm{dx}$, and $\mathrm{dv} = {e}^{x} \mathrm{dx}$, so $v = {e}^{x}$.

Thus we see that the original integral equals:

$= \left[{e}^{x} \left({\sin}^{-} 1 x\right) - \int {e}^{x} / \sqrt{1 - {x}^{2}} \mathrm{dx}\right] + \int {e}^{x} / \sqrt{1 - {x}^{2}} \mathrm{dx}$

The integral $\int {e}^{x} / \sqrt{1 - {x}^{2}} \mathrm{dx}$ will cancel with itself, leaving just

$= {e}^{x} \left({\sin}^{-} 1 x\right)$

$= {e}^{x} \left({\sin}^{-} 1 x\right) + C$

Jul 28, 2016

${e}^{x} \left({\sin}^{-} 1 x\right) + C$

#### Explanation:

There is a pattern surrounding integrals involving ${e}^{x}$, in cases like this problem (and all the others surrounding it on the page).

Since:

$\frac{d}{\mathrm{dx}} {e}^{x} f \left(x\right) = {e}^{x} f \left(x\right) + {e}^{x} f ' \left(x\right) = {e}^{x} \left[f \left(x\right) + f ' \left(x\right)\right]$

Then:

$\int {e}^{x} \left[f \left(x\right) + f ' \left(x\right)\right] \mathrm{dx} = {e}^{x} f \left(x\right) + C$

This is the case for the given problem, which simplifies to be:

$\int {e}^{x} \left[{\sin}^{-} 1 x + \frac{1}{\sqrt{1 - {x}^{2}}}\right] \mathrm{dx}$

If $f \left(x\right) = {\sin}^{-} 1 x$, then $f ' \left(x\right) = \frac{1}{\sqrt{1 - {x}^{2}}}$.

Thus the integral equals

$= {e}^{x} f \left(x\right) + C = {e}^{x} \left({\sin}^{-} 1 x\right) + C$